A FAMILY OF COMPACTIFIED JACOBIANS SATISFIES AUTODUALITY
BECAUSE IT HAS RATIONAL SINGULARITIES
JESSE LEO KASS
A BSTRACT. We prove that the compactified jacobian of a nodal curve satisfies autoduality.
We establish this result by proving a comparison theorem that relates the family of Picard
schemes associated to a family of compactified jacobians to the Néron model, a result of
independent interest. In our proof, a key fact is that the total space of a suitable family of
compactified jacobians has rational singularities.
We prove that the compactified jacobian of a nodal curve satisfies autoduality. We
establish this result by proving a comparison theorem that relates the family of Picard
schemes associated to a family of compactified jacobians to the Néron model, a result of
independent interest. In our proof, a key fact is that the total space of a suitable family of
compactified jacobians has rational singularities.
The classical statement of autoduality is a statement about a smooth projective curve
X0 defined over a field k. Associated to X0 is its jacobian variety J0 . If L0 is a degree -1 line
bundle on X0 , then the rule x0 7→ L0 (x0 ) defines a morphism α0 : X0 → J0 , and the pullback
morphism
(1)
α∗0 : Pic0 (J0 /k) → J0 ,
M0 7→ α∗0 (M0 )
from the Picard scheme of J0 to the jacobian is an isomorphism. The jacobian J0 is an
abelian variety, and the Picard scheme Pic0 (J0 /k) is the dual abelian variety, so the fact
that α∗0 is an isomorphism implies that J0 is self-dual — that the autoduality theorem
holds.
Here we prove that the autoduality theorem holds when X0 is a nodal curve. For such
a X0 , there are two analogues of the jacobian: the generalized jacobian J00 and the compactified jacobian J0 . The generalized jacobian is the moduli space of multidegree 0 line
bundles, and the compactified jacobian is the moduli space of degree 0 rank 1, torsionfree sheaves, which are required to satisfy a semistability condition when X0 is reducible.
In fact, when X0 is reducible, there are many choices of semistability conditions and thus
many compactified jacobians. We discuss this topic below at the end of Section I.
The compactified jacobian J0 is always a (possibly reducible) projective variety, so we
can form the Picard scheme Pic0 (J0 /k) and ask if there is an isomorphism between this
scheme and the generalized jacobian analogous to the isomorphism in Equation (1). We
prove the answer is “Yes”:
Main Theorem (Autoduality). The autoduality theorem holds for the compactified jacobian J0 .
2010 Mathematics Subject Classification. Primary 14H40; Secondary 14K30, 14D20.
1
This is Corollary 6 below.
One consequence of the Main Theorem is that Pic0 (J0 /k) depends only on the curve X0 ,
rather than on the compactified jacobian J0 . Recall J0 depends on a choice of semistability
condition, and different choices produce different schemes. For example, when X0 equals
two rational curves meeting in 3 nodes, one choice produces a J0 with two irreducible
components, while another produces a J0 with three irreducible components. (See [OS79,
Example 13.1(3)].)
This autoduality theorem is new when J0 is coarse, a condition discussed in Section I.
The theorem was known when J0 is fine by work we now review. When X0 is irreducible,
the theorem was proven by Esteves–Gagné–Kleiman [EGK02, Theorem (Autoduality),
pages 5-6]. This result was extended by Esteves–Rocha [ER13, pages 414-415] to treelike curves and by Melo–Rapagnetta–Viviani [MRV12a, Theorem C] to arbitrary nodal
curves. These authors prove results for curves with worse than nodal singularities, and
their work has been generalized in various way, e.g. to curves with planar singularities
[Ari11] and to results about the compactified Picard scheme of J0 [EK05, Ari13, MRV12b].
For fine compactified jacobians, the proof of autoduality we give here, which was inspired by [BLR90, Theorem 1, Section 9.7], is different from previous proofs. Given X0 and
J0 , we realize J0 as the special fiber of a family J/S over S = Spec(k[[t]]) that is associated
to a family of curves X/S s.t. the total space X is a regular scheme.
We compare the family Pic0 (J/S)/S with the Néron model of its generic fiber. (The
Néron model is an extension of the generic fiber to a S-scheme that satisfies a universal
mapping property.) Thus pick a resolution of singularities β : eJ → J and consider the
pullback homomorphism β∗ : Pic0 (J/S) → Pic0 (eJ/S). In Proposition 1, we prove J has
rational singularities, and this implies the differential of β∗ — and hence β∗ itself — is an
isomorphism. A theorem of Pépin states that Pic0 (eJ/S) is the identity component of the
Néron model, so we conclude that:
Theorem (Néron Comparison). Pic0 (J/S)/S is the identity component of the Néron model of
its generic fiber.
This theorem is Theorem 4 below, and it immediately implies the Main Theorem because the universal mapping property of the Néron model implies that the classical autoduality isomorphism of the generic fiber extends over all of S.
The proof just sketched deduces autoduality from the fact that J0 deforms in a family
J/S s.t. J has rational singularities. By contrast, in [EGK02] the result is deduced from a
description of J0 coming from the presentation scheme, in [ER13] from autoduality for
irreducible curves, and in [MRV12a] from the computation of the cohomology of a universal family of sheaves, a computation done by putting J0 into a suitable miniversal
family.
A word about the characteristic. In this paper we assume:
Assumption. the base field k has characteristic 0.
2
We need to make this assumption because we make use of properties of rational singularities. There is a well-developed theory of rational singularities in characteristic zero,
but not in positive characteristic (except for the case of surface singularities). To extend
the proof of the main results of this paper to allow k to have positive characteristic, it
would be enough to prove that the total space J of a family of compactified jacobians
admits a rational resolution and to prove that Corollary 3 remains valid.
I. A
REVIEW OF COMPACTIFIED JACOBIANS
Here we recall the definition of the compactified jacobian and related objects. We begin
by fixing notation. Let k be a field of characteristic zero. A curve X0 /Spec(k) is a k-scheme
that is geometrically connected, geometrically reduced, 1-dimensional, and proper over
k. We set g := 1 − χ(X0 , OX0 ) equal to the arithmetic genus. When k = k is algebraically
bX ,x of X0 at a point
closed, we say that X0 /Spec(k) is nodal if the completed local ring O
0 0
not lying in the k-smooth locus is isomorphic to k[[x, y]]/(xy). In general, we say that
X0 /Spec(k) is nodal if X0 ⊗k k is nodal. A family of curves over a scheme T is a T -scheme
X/T that is proper and flat over T and s.t. the fibers of X → T are curves. If the fibers are
nodal curves, then we say X/T is a family of nodal curves. We say that a family of curves
X/S is regular if X is a regular scheme (i.e. at every closed point the Zariski tangent space
has the minimal dimension 2).
If X0 /Spec(k) is a curve, then we say that a line bundle L0 has multidegree zero if for all
irreducible components Y0 of X0 ⊗k k the restriction of L0 ⊗k k to Y0 has degree 0 (i.e. the
Euler characteristic of the restriction equals χ(Y0 , OY0 )). A OX -module L on a family of
curves X/T is called a family of multidegree 0 line bundles if L is a line bundle s.t. the
restriction to any fiber of X → T has multidegree 0. The étale sheafification of the functor
that assigns to a k-scheme T the set of isomorphism classes of families of multidegree 0
line bundles on X0 ×k T is representable by a k-scheme J00 called the generalized jacobian
of X0 /Spec(k). The generalized jacobian is quasi-projective and smooth over k.
Also associated to X0 /Spec(k) is its compactified jacobian. Suppose that we are given an
ample line bundle A0 on X0 . We say that a rank 1, torsion-free sheaf I0 on X0 is semistable
(resp. stable) with respect to A0 if the slope µ(I0 ) := χ(X0 , I0 )/ deg(A0 ) satisfies µ(J0 ) ≤
µ(I0 ) (resp. µ(J0 ) < µ(I0 )) for all nonzero subsheaves J0 ⊂ I0 . If X/T is a family of curves
with family of ample line bundles A (i.e. A is a line bundle on X s.t. the restriction to
every fiber of X → T is ample), then a family of rank 1, torsion-free sheaves on X/T
that is semistable with respect to A is a OT -flat, finitely presented OX -module I s.t. the
restriction to every fiber of X → T is a rank 1, torsion-free sheaf semistable with respect to
the restriction of A.
The compactified jacobian J0 /Spec(k) associated to a curve X0 /Spec(k) and an ample
line bundle A0 is the k-scheme that universally corepresents the functor that assigns to a
k-scheme T the set of isomorphism classes of families of rank 1, torsion-free sheaves on
X0 ×k T that are semistable with respect to A0 ⊗k OT . The compactified jacobian exists
and is projective over k by [Sim94, Theorem 1.21] (Note: the moduli space described in
loc. cite includes pure sheaves that fail to have rank 1, and J0 is a connected component
3
of this larger moduli space; when stability coincides with semistability, this is shown in
[Kas13, Section 4.2], and the semistable case can be treated by applying the argument in
loc. cite to a suitable Quot scheme). We call J0 the compactified jacobian. We say that
J0 is a fine compactified jacobian if every degree 0 semistable rank 1, torsion-free sheaf is
stable. Otherwise we say that J0 is coarse.
Suppose now that k is the residue field of a discrete valuation ring R with field of fractions K. Given a family of curves X/S we define the associated family of generalized
jacobians J0 /S to be the k-scheme that represents the étale sheafification of the functor
assigning to a S-scheme T the set of isomorphism classes of families of multidegree zero
line bundles on X ×S T . The family of generalized jacobians J0 /S exists as a S-scheme
that is smooth and quasi-projective over S. The fibers of J0 → S are the generalized jacobians of the fibers of X → S (because the formation of the functor J0 represents commutes
with fiber products). Given a family of ample line bundles A on X/S, the functor that
assigns to a S-scheme T the set of isomorphism classes of families of rank 1, torsion-free
sheaves on X ×S T that are semistable with respect to A is universally corepresented by a
S-scheme J/S that is projective over S. We call J/S the family of compactified jacobians
associated to X/S and A. The fibers of J → S are the compactified jacobians of the fibers
of X → S (because the formation of the functor J universally corepresents commutes with
fiber products).
A word about the compactified jacobians we study in this paper. They are more properly called Simpson compactified jacobians or slope semistable compactified jacobians.
Other compactified jacobians have been constructed (see e.g. [Kas13] for a brief survey).
We restrict our attention to slope semistable compactified jacobians only to keep our review of compactified jacobians short. The author expects that the results of this paper
remain valid for the other compactified jacobians that have been constructed. Indeed,
the key results we use about J are Propositions 1 and 2, and the proofs of these propositions remain valid for any family of compactified jacobians that is a moduli space of
rank 1, torsion-free sheaves that is either fine or is constructed using Geometric Invariant
Theory.
II. T HE SINGULARITIES OF J
Here we prove some results about the singularities of a family of compactified jacobians
associated to a family of nodal curves. We use these results in Section III; Corollary 3 of
this section is used to show that the family of Picard schemes associated to a family of
compactified jacobians is well-behaved, and Proposition 1 is used to prove Theorem 4,
the Néron Comparison Theorem.
In this section, S is the spectrum of a fixed discrete valuation ring R with residue field k,
X/S is a regular family of nodal curves, and J/S is a family J/S of compactified Jacobians
associated to X/S. We assume k has characteristic zero.
4
We prove the main results of this section using a local description of J/S obtained from
deformation theory. When k = k is algebraically closed and J/S is a family of fine compactified jacobians, the completed local ring of J at a closed point x0 ∈ J can be described
as:
∼ R[[u
b =
b 1 , v1 , . . . , un , vn , w1 , . . . , wm ]]/(u1 v1 − π, . . . , un vn − π)
(2)
O
J,x
for some uniformizer π ∈ R and some integers n, m ∈ N. This is [Kas09, Lemma 6.2], a
result proven using the techniques used in [CMKV12].
When J/S is a family of coarse compactified jacobians, the Luna Slice argument used in
the loc. cite shows that there is a multiplicative torus Gkm acting on the ring appearing on
b .
the right-hand side of Equation (2) s.t. the torus invariant subring is isomorphic to O
J,x
Using this result, we prove:
Proposition 1. J has rational singularities, and J → S is flat.
Proof. We can assume k = k because it is enough to prove the result after passing from
R to its strict henselization Rsh . With this assumption, suppose first that J/S is a family of fine compactified jacobian. The morphism J → S is flat because the ring appearing in Equation (2) is the quotient of a power series by elements whose images in
b 1 , v1 , . . . , wm ]]/(π) form a regular sequence [Mat89, Corollary to Theorem 22.5]. To
R[[u
b is isomorphic to the completion of
see that J has rational singularities, observe that O
J,x0
k[u1 , v1 , . . . , un , vn , w1 , . . . , wm ]/(u1 v1 −u2 v2 , . . . , u1 v1 −un vn ), which is the coordinate ring
of an affine toric variety. (An isomorphism is determined by a choice of coefficient field
k ⊂ R.) Since toric varieties have rational singularities, so does J, proving the proposition
when J is fine.
When J/S is coarse, the argument just given shows that, if x0 ∈ J is a closed point, then
b
OJ,x0 is the torus invariant subring of a ring that is R-flat and has rational singularities. In
b has rational singularities by [Bou87, Corollary, page 66] and is flat over R
particular, O
J,x0
as it is a direct summand of a flat module.
From Equation (2), we deduce that when k is algebraically closed and J0 is a fine compactified jacobian, the completed local ring of J0 at a closed point x0 ∈ J0 is:
(3)
∼
b
O
J,x0 = k[[u1 , v1 , . . . , un , vn , w1 , . . . , wm ]]/(u1 v1 , . . . , un vn ).
b is isomorphic to the subring of the ring appearing on the rightWhen J0 is coarse, O
J,x
hand side of Equation (3) for some action of a multiplicative torus Gkm . We use these
descriptions to prove:
Proposition 2. J0 has Du Bois singularities.
Proof. We can assume k = k. When J0 is fine, Equation (3) shows that the completed local
ring of J0 at a closed point x is a completed product of double normal crossing singularity
rings and power series rings, and such a completed product is Du Bois by [Doh08, Example 3.3, Theorem 3.9]. When J0 is coarse, the completed local ring of J0 at a closed point is
5
the torus invariant subring of a Du Bois local ring and hence is itself Du Bois by [Kov99,
Corollary 2.4] (the left inverse hypothesis is satisfied because the torus invariant subring
is a direct summand).
From Proposition 2, we deduce:
Corollary 3. The higher
direct image Ri p∗ OJ of OJ under the projection p : J → S is a locally free
OS -module of rank gi , and its formation commutes with arbitrary base chance.
Proof. The fibers of J → S have Du Bois singularities, so the result is [DB81, Théorème 4.6].
III. C OMPARISON WITH THE N ÉRON MODEL
Here we prove a comparison theorem relating a family of Picard schemes to the Néron
model of its generic fiber. As in Section II we fix the spectrum S of a dvr R with residue
field k and field of fractions K, a regular family of nodal curves X/S, and a family J/S of
compactified Jacobians associated to X/S. We assume k has characteristic zero.
The family of Picard schemes Pic(J/S)/S associated to J/S is the S-scheme that represents the fppf sheafification of the functor that assigns to a S-scheme T the set Pic(JT )
of isomorphism classes of line bundles on J ×S T . The family of Picard schemes exists
as a (possibly nonseparated) S-group space that is locally of finite presentation over S.
Indeed, because the formation of the pushforward p∗ OJ by p : J → S commutes with base
change, this representability result is [Ray70, (1.5)]. The sheaf R1 p∗ OJ is locally free and
its formation also commutes with base change (Corollary 3), so by [Kle05, Corollary 5.14,
Proposition 5.20] Pic(J/S)/S contains the identity component Pic0 (J/S)/S, an open Ssubgroup scheme that is of finite type and smooth over S and has the property that the
fibers of Pic0 (J/S) → S are the identity components of the fibers of Pic(J/S) → S. We
denote the identity component by P0 /S = P0 (J/S) and define P/S = P(J/S)/S to be the
closure of the generic fiber PK0 in Pic(J/S). Because P0 is smooth over S, it is contained in
P.
We compare P0 to the Néron model of its generic fiber PK . The Néron model N/S of
PK is a S-scheme that is smooth over S, contains PK as the generic fiber, and satisfies the
Néron mapping property; that is, for every smooth morphism T → S the natural map
(4)
HomS (T, N) → HomK (TK , PK )
is bijective. By a theorem of Néron N/S exists and is separated and of finite type over
S [BLR90, Corollary 2, Section 9.7]. The identity component N0 /S is defined to be the
complement of the connected components of the special fiber Nk that do not contain the
identity element. By construction N0 is an open S-group subscheme of N s.t. the fibers of
N0 → S are connected.
The identity morphism idK : PK → PK extends uniquely to a S-morphism
(5)
P0 → N0 ,
6
and we prove:
Theorem 4 (Néron Comparison). The morphism (5) is an isomorphism.
Proof. We prove this theorem by choosing a regular S-model eJ/S of J/S, using a theorem
of Pépin to relate the family of Picard schemes of eJ/S to the Néron model, and then using
the rational singularities result, Proposition 1, to show that eJ/S and J/S have isomorphic
families of Picard schemes.
Let p : J → S be the structure morphism and β : eJ → J a resolution of singularities with
exceptional locus contained in the singular locus. Because J has rational singularities
(Proposition 1), the higher direct images Rj β∗ OeJ vanish for j > 0 and the direct image
satisfies β∗ OeJ = OJ . The Leray spectral sequence Ri p∗ ◦ Rj β∗ OeJ ⇒ Ri+j (p ◦ β)∗ OeJ thus
degenerates at the E2 page, so the natural homomorphisms
Ri p∗ OJ = Ri p∗ ◦ R0 β∗ OeJ → Ri (p ◦ β)∗ OeJ
are isomorphisms. In particular, the direct image R1 (p ◦ β)∗ OeJ is locally free of rank g and
its formation commutes with base change (Proposition 2).
This shows that the hypothesis of [Ray70, (1.5)] holds, so the family of Picard schemes
Pic(eJ/S)/S exists as a S-group space that is locally of finite presentation over S. Define
P0 (eJ/S)/S and P(eJ/S)/S in analogy with P0 (J/S)/S and P(J/S)/S. The identity component
of the S-group smoothening of P(eJ/S) is isomorphic to N0 by [Pép13, Proposition 10.3].
In fact, it is equal to its S-group smoothening. Indeed, P(eJ/S) is smooth over S because
it is flat (as its generic fiber is dense) and the fibers of P(eJ/S) → S are smooth (by [Kle05,
Corollay 5.15] and the fact that R1 (p ◦ β)∗ OeJ satisfies the analogue of Proposition 2), so
Pépin’s result asserts that the morphism
P0 (J/S) → N0
extending the identity map is an isomorphism. Thus to prove the theorem, it is enough
to show that
β∗ : P0 (J/S) → P0 (eJ/S),
M 7→ β∗ (M)
is an isomorphism.
Consider the map β∗ induces on Lie algebras. The map on Lie algebras is the natural
homomorphism
R1 p∗ OJ → R1 (p ◦ β)∗ OeJ ,
and we already observed that this is an isomorphism.
We conclude that β∗ is étale. In particular, β∗ has finite fibers. The morphism is also
birational (β∗K is an isomorphism), so β∗ must be an open immersion by Zariski’s main
theorem. Because the fibers of P0 (eJ/S) → S are connected, the only open S-subgroup
scheme of P0 (eJ/S) is P0 (eJ/S), and so β∗ is an isomorphism.
7
Remark 5. The Néron Comparison Theorem is sharp in the following sense. The theorem shows that the identity component of the Néron model is isomorphic to an open Ssubgroup scheme of Pic(J/S), and one can ask if there is a larger open subgroup scheme
that is isomorphic to the Néron model. Without additional hypotheses, no such larger
subgroup scheme exists. We demonstrate this with the following example.
Let S equal Spec(C[t](t) ) (the localization of C[t] at (t)), X the minimal regular model
of Spec(R[x, y]/(y2 − x3 − x2 − t2 )), and J/S the family of degree 0 compactified jacobians
associated to the canonical polarization ωX/S . Then X/S is a family of genus 1 curves
whose special fiber X0 consists of two rational curves meeting in two nodes, and J/S is
a family of genus 1 curves whose special fiber J0 is irreducible. Since J/S is a family of
curves, Pic0 (J/S) is flat over S and thus Pic0 (J/S) is equal to the closure of its generic fiber
in Pic(J/S). We can conclude that Pic0 (J/S)/S is the largest subgroup scheme of Pic(J/S)
that contains the identity component Pic0 (J/S) and is isomorphic to an open subgroup
scheme of the Néron model (for any open scheme of the Néron model has dense generic
fiber by S-smoothness).
The identity component Pic0 (J/S) is not, however, the Néron model of its generic fiber
because the Néron model has disconnected special fiber. (The elliptic curve JK has reduction type I2 in Kodaira’s classification [Sil94, Theorem 8.2].) The theorems [Pép13,
Théoréme 9.3] and [Ray70, Théoréme 8.1.4] suggest that one should not ask for an open
subgroup of Pic(J/S) isomorphic to the Néron model, but rather for an open subgroup
scheme whose maximal separated quotient is isomorphic to the Néron model. In the
example just discussed, Pic(J/S) is separated, so again no such open subgroup scheme
exists.
IV. A UTODUALITY
Here we use Theorem 4, the Néron Comparison Theorem, to prove that the compactified jacobian of a nodal curve satisfies autoduality. In other words, if k is a field of characteristic zero, X0 /Spec(k) a nodal curve, and J0 /Spec(k) a compactified jacobian associated
∼ Pic0 (J0 /k) between the generalized jacobian
to X0 , then we construct an isomorphism J00 =
and the identity component of the Picard scheme of J0 — the autoduality isomorphism.
This autoduality isomorphism is realized by the Abel map when the Abel map is defined. Recall that the Abel map of a non-singular curve X0 is defined as follows. If L0 is a
line bundle on X0 of degree −1, then the rule
(6)
x0 7→ L0 (x0 )
defines a morphism α0 = αL0 : X0 → J0 that is the Abel map (associated to L0 ).
What if X0 is a nodal curve? When X0 is irreducible, Equation (6) defines a morphism
X0 → J0 into the compactified jacobian by [EGK02, 2.2], but when X0 is reducible, the
equation can fail to define a morphism because L0 (x0 ) can fail to be semistable. When
L0 (x0 ) is semistable for all x0 , we say that the morphism defined by Equation (6) is the
Abel map associated to L0 .
8
The problem of constructing a L0 s.t. L0 (x0 ) is always semistable (i.e. of constructing an
Abel map) is nontrivial. This and related problems are studied in [Cap07, CE07, CCE08,
CP10], and we direct the reader to those papers for results about the existence of an Abel
map for a reducible curve.
We now state and prove the autoduality result.
∼ J0 , and this isomorphism is pullback by an Abel map
Corollary 6 (Autoduality). Pic0 (J0 /k) =
0
when an Abel map is defined.
Proof. We deduce this result by deforming J0 to the jacobian JK of a non-singular curve and
then using Theorem 4 to argue that the classical autoduality isomorphism for JK extends
to an autoduality isomorphism for J0 .
We can realize J0 as the special fiber of a family of compactified jacobians J/S associated
to a family of curves X/S over the spectrum S = Spec(k[[t]]) of a power series ring s.t. the
total space X is regular. Indeed, by [Bak08, Theorem B.2] (or [Win74, Theorem 2.5]) we
can realize X0 as the special fiber of a flat family of curves X/S s.t. X is regular. Now
suppose J0 is the compactified jacobian parameterizing sheaves semistable with respect
to the ample line bundle A0 . We can realize A0 as the special fiber of a family of ample
line bundles A on X as the relevant family of Picard schemes is smooth over S. The family
of compactified jacobians J/S associated to A has the desired properties.
There are S-isomorphisms
(7)
(8)
∼ (N∨ )0 by [BLR90, Theorem 1, page 286]
J0 =
∼ N0 by Theorem 4
Pic0 (J/S) =
uniquely determined by the requirement that they restrict to the identity on the generic
fiber. Here N∨ is the Néron model of JK and N is the Néron model of PK .
∼ Pic0 (JK /K) extends to an isomorphism
The autoduality isomorphism JK =
(9)
∼N
N∨ =
by the Néron mapping property. The isomorphism (9) restricts to an isomorphism between identity components, and the composition of this restriction with the isomorphisms
∼ J0 . We define this to be the autoduality iso(7) and (8) is an isomorphism Pic0 (J0 /k) =
0
morphism, proving the first part of the corollary.
To complete the proof, we need to show that if X0 admits an Abel map α0 , then the
autoduality isomorphism is the pullback homomorphism α∗0 . Thus suppose that α0 : X0 →
J0 is the morphism x 7→ L0 (x) for some degree -1 line bundle L0 on X0 . We can extend L0 to
a line bundle L on X (again by smoothness of the appropriate relative Picard scheme) and
thus extend α0 to the morphism α : X → J defined by x 7→ L(x). The pullback morphism
α∗ : Pic0 (J/S) → J0 and the autoduality isomorphism agree over the generic fiber Spec(K)
(by classical autoduality), so they must be equal over S. We deduce the desired result by
restricting to the special fiber.
9
Acknowledgements. This work was begun while the author was a Wissenschaftlicher
Mitarbeiter at the Institut für Algebraische Geometrie, Leibniz Universität Hannover.
During the writing of this paper the author received support from a AMS–Simons Travel
Grant.
TO BE ADDED AFTER THE ARTICLE HAS BEEN REFEREED.
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D EPT. OF M ATHEMATICS , U NIVERSITY OF S OUTH C AROLINA , C OLUMBIA SC
E-mail address: [email protected]
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